2,241 research outputs found
Degree-degree correlations in random graphs with heavy-tailed degrees
Mixing patterns in large self-organizing networks, such as the Internet, the
World Wide Web, social and biological networks are often characterized by
degree-degree {dependencies} between neighbouring nodes. One of the problems
with the commonly used Pearson's correlation coefficient (termed as the
assortativity coefficient) is that {in disassortative networks its magnitude
decreases} with the network size. This makes it impossible to compare mixing
patterns, for example, in two web crawls of different size.
We start with a simple model of two heavy-tailed highly correlated random
variable and , and show that the sample correlation coefficient
converges in distribution either to a proper random variable on , or to
zero, and if then the limit is non-negative. We next show that it is
non-negative in the large graph limit when the degree distribution has an
infinite third moment. We consider the alternative degree-degree dependency
measure, based on the Spearman's rho, and prove that it converges to an
appropriate limit under very general conditions. We verify that these
conditions hold in common network models, such as configuration model and
Preferential Attachment model. We conclude that rank correlations provide a
suitable and informative method for uncovering network mixing patterns
Convergence of rank based degree-degree correlations in random directed networks
We introduce, and analyze, three measures for degree-degree dependencies,
also called degree assortativity, in directed random graphs, based on
Spearman's rho and Kendall's tau. We proof statistical consistency of these
measures in general random graphs and show that the directed configuration
model can serve as a null model for our degree-degree dependency measures.
Based on these results we argue that the measures we introduce should be
preferred over Pearson's correlation coefficients, when studying degree-degree
dependencies, since the latter has several issues in the case of large networks
with scale-free degree distributions
Phase transitions for scaling of structural correlations in directed networks
Analysis of degree-degree dependencies in complex networks, and their impact
on processes on networks requires null models, i.e. models that generate
uncorrelated scale-free networks. Most models to date however show structural
negative dependencies, caused by finite size effects. We analyze the behavior
of these structural negative degree-degree dependencies, using rank based
correlation measures, in the directed Erased Configuration Model. We obtain
expressions for the scaling as a function of the exponents of the
distributions. Moreover, we show that this scaling undergoes a phase
transition, where one region exhibits scaling related to the natural cut-off of
the network while another region has scaling similar to the structural cut-off
for uncorrelated networks. By establishing the speed of convergence of these
structural dependencies we are able to asses statistical significance of
degree-degree dependencies on finite complex networks when compared to networks
generated by the directed Erased Configuration Model
Average nearest neighbor degrees in scale-free networks
The average nearest neighbor degree (ANND) of a node of degree is widely
used to measure dependencies between degrees of neighbor nodes in a network. We
formally analyze ANND in undirected random graphs when the graph size tends to
infinity. The limiting behavior of ANND depends on the variance of the degree
distribution. When the variance is finite, the ANND has a deterministic limit.
When the variance is infinite, the ANND scales with the size of the graph, and
we prove a corresponding central limit theorem in the configuration model (CM,
a network with random connections). As ANND proved uninformative in the
infinite variance scenario, we propose an alternative measure, the average
nearest neighbor rank (ANNR). We prove that ANNR converges to a deterministic
function whenever the degree distribution has finite mean. We then consider the
erased configuration model (ECM), where self-loops and multiple edges are
removed, and investigate the well-known `structural negative correlations', or
`finite-size effects', that arise in simple graphs, such as ECM, because large
nodes can only have a limited number of large neighbors. Interestingly, we
prove that for any fixed , ANNR in ECM converges to the same limit as in CM.
However, numerical experiments show that finite-size effects occur when
scales with
Provable and practical approximations for the degree distribution using sublinear graph samples
The degree distribution is one of the most fundamental properties used in the
analysis of massive graphs. There is a large literature on graph sampling,
where the goal is to estimate properties (especially the degree distribution)
of a large graph through a small, random sample. The degree distribution
estimation poses a significant challenge, due to its heavy-tailed nature and
the large variance in degrees.
We design a new algorithm, SADDLES, for this problem, using recent
mathematical techniques from the field of sublinear algorithms. The SADDLES
algorithm gives provably accurate outputs for all values of the degree
distribution. For the analysis, we define two fatness measures of the degree
distribution, called the -index and the -index. We prove that SADDLES is
sublinear in the graph size when these indices are large. A corollary of this
result is a provably sublinear algorithm for any degree distribution bounded
below by a power law.
We deploy our new algorithm on a variety of real datasets and demonstrate its
excellent empirical behavior. In all instances, we get extremely accurate
approximations for all values in the degree distribution by observing at most
of the vertices. This is a major improvement over the state-of-the-art
sampling algorithms, which typically sample more than of the vertices to
give comparable results. We also observe that the and -indices of real
graphs are large, validating our theoretical analysis.Comment: Longer version of the WWW 2018 submissio
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